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This is not homework, but is a problem I am asking myself.

A long rod with known elastic material properties floats in flat space - suspended and kept in place by a number of massless ropes.

Now we apply a static gravitational field and we wait long enough so that everything in the situation becomes static again. (Thus we disregard transient effects)

(1) Is there a simple way to quantify how the bending of the rod and the bending of the space at the same position are related? Is the bending of the rod always larger?

(2) Same question about stretching...

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2 Answers 2

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I'll give this a try. Feel free to butt-in!

Note that the rod is embedded within space. The bending of the rod is due to space itself being bent, hence the rod experiences no mechanical stress. As such, the degree of the rod bending has nothing to do with its elastic properties. Bending of space does not induce mechanical stress.

I'd venture to say that, to quantify the bending of the rod is to quantify the bending of space and vice versa. I can't see that matter have some inertia-like property against bending in space, thus the bending is direct in the sense that; the rod is bent to the same degree as space is and vice versa.

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  • $\begingroup$ Tides show that the bending of bodies is usually much larger than the bending of space. They are never equal. $\endgroup$
    – Hans973
    Feb 21, 2015 at 4:24
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Is there a way to quantify how the bending of the rod and the bending of the space at the same position are related?

There is a way to quantify how the deviation of spacetime from being "flat" (in the region containing the given rod) and the deviation of the (eventual, static) "shape" of the rod constituents with respect to each other, which is (eventually) attained or approached if not all constituents of the rod are individually rigidly supported by some external means:

It is (apparently) called the "(static) material properties" of that rod.

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  • $\begingroup$ This does not answer the question, does it? $\endgroup$
    – Hans973
    Feb 22, 2015 at 10:25
  • $\begingroup$ Hans973: "This does not answer the question, does it?" -- Correct. My response is meant to reject your OP question, as not properly and in good conscience answerable; and to sketch how to ask such a question instead, in the given context (namely: "Is there a way to quantify material properties?"). Hope this helps. $\endgroup$
    – user12262
    Feb 22, 2015 at 11:47
  • $\begingroup$ Bending of rods can be quantified with geometric quantities and is done so since hundreds of years. Bending of space can be quantified with general relativity, which is 100 years old. The question asks about a simple way to determine the ratio of the two. If a rod is not ok, take a plate, or a ring, or whatever else you like. $\endgroup$
    – Hans973
    Feb 23, 2015 at 12:37
  • $\begingroup$ Hans973: "Bending of rods can be quantified with geometric quantities" -- Well, certainly the shape of some "bunch of (distinct, identifiable) constituents" is a geometric property (of that "bunch") to be quantified. -- "done so since hundreds of years." -- More or less, sure. "Bending of space can be quantified with GR, which is 100 years old." -- "Bending of spacetime", right. "The question asks about a simple way to determine the ratio of the two." -- This is trivial: just take that ratio; trial by trial. However, the Orig.Post instead presumed this ratio outright. $\endgroup$
    – user12262
    Feb 23, 2015 at 18:59

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